Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 264347, 23 pages
Research Article

On an Inequality of H. G. Hardy

1Abdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan
2Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovića 28a, 10000 Zagreb, Croatia

Received 18 June 2010; Accepted 16 October 2010

Academic Editor: Q. Lan

Copyright © 2010 Sajid Iqbal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We state, prove, and discuss new general inequality for convex and increasing functions. As a special case of that general result, we obtain new fractional inequalities involving fractional integrals and derivatives of Riemann-Liouville type. Consequently, we get the inequality of H. G. Hardy from 1918. We also obtain new results involving fractional derivatives of Canavati and Caputo types as well as fractional integrals of a function with respect to another function. Finally, we apply our main result to multidimensional settings to obtain new results involving mixed Riemann-Liouville fractional integrals.